Question

A value of \( \theta \) lying between \( 0 \) and \( \pi / 2 \) and satisfying \[ \begin{vmatrix} 1 + \sin^2 \theta & \cos^2 \theta & 4 \sin 4\theta \sin^2 \theta & 1 + \cos^2 \theta & 4 \sin 4\theta \sin^2 \theta & \cos^2 \theta & 1 + 4 \sin 4\theta \end{vmatrix} = 0 \] is:

• A. \( \frac{5\pi}{24} \)
• B. \( \frac{7\pi}{24} \)
• C. \( \frac{\pi}{8} \)
• D. \( \frac{3\pi}{8} \)

Hint:

When a determinant contains cyclic or repeating trigonometric functions across rows or columns, look for column operations like \(C_1 + C_2\) to exploit the standard identity \(\sin^2\theta + \cos^2\theta = 1\). This instantly reduces variables to constants!

Answer 1

The Correct Option is B

To solve the given problem, we need to evaluate the determinant of the matrix provided and find a value of \( \theta \) between \( 0 \) and \( \pi/2 \) that makes the determinant zero.

Step 1: Analyze the Given Determinant

The given expression involves a determinant with components:

• Diagonals and off-diagonal elements are combinations of trigonometric expressions in \(\theta\). These involve \(\sin^2 \theta\), \(\cos^2 \theta\), and \(\sin 4\theta \sin^2 \theta\).
• Use the identity: \(\sin^2 \theta + \cos^2 \theta = 1\)to simplify some expressions.

Step 2: Evaluate the Condition for Determinant to be Zero

Given: 

\[\begin{vmatrix} 1 + \sin^2 \theta & \cos^2 \theta & 4 \sin 4\theta \sin^2 \theta \\ 1 + \cos^2 \theta & 4 \sin 4\theta \sin^2 \theta & \cos^2 \theta \\ 1 + 4 \sin 4\theta \end{vmatrix} = 0\]

The determinant simplifies to a condition involving these trigonometric terms. Simplifying this further reveals that the root of the expression depends on the periodicity and range of these trigonometric functions.

Step 3: Test the Given Options

1. \(\theta = \frac{5\pi}{24}\): Evaluate the trigonometric expressions for this value. Check if determinant simplifies to zero.
2. \(\theta = \frac{7\pi}{24}\): Evaluate similarly.
3. \(\theta = \frac{\pi}{8}\): Evaluate similarly.
4. \(\theta = \frac{3\pi}{8}\): Evaluate similarly.

On evaluating these options, the value of \(\theta = \frac{7\pi}{24}\) results in the determinant simplifying to zero.

Conclusion

The value of \(\theta\) between \(0\) and \(\pi/2\) which satisfies the condition is \(\frac{7\pi}{24}\).